3.1.2 \(\int (a+b \coth ^2(c+d x))^4 \, dx\) [2]

Optimal. Leaf size=110 \[ (a+b)^4 x-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {b^3 (4 a+b) \coth ^5(c+d x)}{5 d}-\frac {b^4 \coth ^7(c+d x)}{7 d} \]

[Out]

(a+b)^4*x-b*(2*a+b)*(2*a^2+2*a*b+b^2)*coth(d*x+c)/d-1/3*b^2*(6*a^2+4*a*b+b^2)*coth(d*x+c)^3/d-1/5*b^3*(4*a+b)*
coth(d*x+c)^5/d-1/7*b^4*coth(d*x+c)^7/d

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Rubi [A]
time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3742, 398, 212} \begin {gather*} -\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^3 (4 a+b) \coth ^5(c+d x)}{5 d}+x (a+b)^4-\frac {b^4 \coth ^7(c+d x)}{7 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Coth[c + d*x]^2)^4,x]

[Out]

(a + b)^4*x - (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Coth[c + d*x])/d - (b^2*(6*a^2 + 4*a*b + b^2)*Coth[c + d*x]^3
)/(3*d) - (b^3*(4*a + b)*Coth[c + d*x]^5)/(5*d) - (b^4*Coth[c + d*x]^7)/(7*d)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3742

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[c*(ff/f), Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

\begin {align*} \int \left (a+b \coth ^2(c+d x)\right )^4 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^4}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-b (2 a+b) \left (2 a^2+2 a b+b^2\right )-b^2 \left (6 a^2+4 a b+b^2\right ) x^2-b^3 (4 a+b) x^4-b^4 x^6+\frac {(a+b)^4}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d}\\ &=-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {b^3 (4 a+b) \coth ^5(c+d x)}{5 d}-\frac {b^4 \coth ^7(c+d x)}{7 d}+\frac {(a+b)^4 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=(a+b)^4 x-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {b^3 (4 a+b) \coth ^5(c+d x)}{5 d}-\frac {b^4 \coth ^7(c+d x)}{7 d}\\ \end {align*}

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Mathematica [A]
time = 1.45, size = 127, normalized size = 1.15 \begin {gather*} -\frac {\coth (c+d x) \left (b \left (105 \left (4 a^3+6 a^2 b+4 a b^2+b^3\right )+35 b \left (6 a^2+4 a b+b^2\right ) \coth ^2(c+d x)+21 b^2 (4 a+b) \coth ^4(c+d x)+15 b^3 \coth ^6(c+d x)\right )-105 (a+b)^4 \tanh ^{-1}\left (\sqrt {\tanh ^2(c+d x)}\right ) \sqrt {\tanh ^2(c+d x)}\right )}{105 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Coth[c + d*x]^2)^4,x]

[Out]

-1/105*(Coth[c + d*x]*(b*(105*(4*a^3 + 6*a^2*b + 4*a*b^2 + b^3) + 35*b*(6*a^2 + 4*a*b + b^2)*Coth[c + d*x]^2 +
21*b^2*(4*a + b)*Coth[c + d*x]^4 + 15*b^3*Coth[c + d*x]^6) - 105*(a + b)^4*ArcTanh[Sqrt[Tanh[c + d*x]^2]]*Sqr
t[Tanh[c + d*x]^2]))/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(213\) vs. \(2(104)=208\).
time = 0.35, size = 214, normalized size = 1.95

method	result	size
derivativedivides	\(\frac {-4 a \,b^{3} \coth \left (d x +c \right )-4 a^{3} b \coth \left (d x +c \right )-6 a^{2} b^{2} \coth \left (d x +c \right )+\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \ln \left (\coth \left (d x +c \right )+1\right )}{2}-\frac {4 a \,b^{3} \left (\coth ^{5}\left (d x +c \right )\right )}{5}-2 a^{2} b^{2} \left (\coth ^{3}\left (d x +c \right )\right )-\frac {4 a \,b^{3} \left (\coth ^{3}\left (d x +c \right )\right )}{3}-b^{4} \coth \left (d x +c \right )-\frac {b^{4} \left (\coth ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \ln \left (\coth \left (d x +c \right )-1\right )}{2}-\frac {b^{4} \left (\coth ^{3}\left (d x +c \right )\right )}{3}-\frac {b^{4} \left (\coth ^{5}\left (d x +c \right )\right )}{5}}{d}\)	\(214\)
default	\(\frac {-4 a \,b^{3} \coth \left (d x +c \right )-4 a^{3} b \coth \left (d x +c \right )-6 a^{2} b^{2} \coth \left (d x +c \right )+\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \ln \left (\coth \left (d x +c \right )+1\right )}{2}-\frac {4 a \,b^{3} \left (\coth ^{5}\left (d x +c \right )\right )}{5}-2 a^{2} b^{2} \left (\coth ^{3}\left (d x +c \right )\right )-\frac {4 a \,b^{3} \left (\coth ^{3}\left (d x +c \right )\right )}{3}-b^{4} \coth \left (d x +c \right )-\frac {b^{4} \left (\coth ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \ln \left (\coth \left (d x +c \right )-1\right )}{2}-\frac {b^{4} \left (\coth ^{3}\left (d x +c \right )\right )}{3}-\frac {b^{4} \left (\coth ^{5}\left (d x +c \right )\right )}{5}}{d}\)	\(214\)
risch	\(x \,a^{4}+4 a^{3} b x +6 a^{2} b^{2} x +4 a \,b^{3} x +b^{4} x -\frac {8 b \left (161 a \,b^{2}+315 a^{2} b \,{\mathrm e}^{12 d x +12 c}+315 a \,b^{2} {\mathrm e}^{12 d x +12 c}-1575 a^{2} b \,{\mathrm e}^{10 d x +10 c}-1260 a \,b^{2} {\mathrm e}^{10 d x +10 c}+2835 a^{2} b \,{\mathrm e}^{4 d x +4 c}-1155 a^{2} b \,{\mathrm e}^{2 d x +2 c}+210 a^{2} b +105 a^{3}+44 b^{3}+2555 a \,b^{2} {\mathrm e}^{8 d x +8 c}-3080 a \,b^{2} {\mathrm e}^{6 d x +6 c}+2121 a \,b^{2} {\mathrm e}^{4 d x +4 c}+3360 a^{2} b \,{\mathrm e}^{8 d x +8 c}-812 a \,b^{2} {\mathrm e}^{2 d x +2 c}-3990 a^{2} b \,{\mathrm e}^{6 d x +6 c}-630 a^{3} {\mathrm e}^{2 d x +2 c}+105 a^{3} {\mathrm e}^{12 d x +12 c}+105 b^{3} {\mathrm e}^{12 d x +12 c}-630 a^{3} {\mathrm e}^{10 d x +10 c}-203 b^{3} {\mathrm e}^{2 d x +2 c}+770 b^{3} {\mathrm e}^{8 d x +8 c}+1575 a^{3} {\mathrm e}^{4 d x +4 c}+609 b^{3} {\mathrm e}^{4 d x +4 c}-315 b^{3} {\mathrm e}^{10 d x +10 c}+1575 a^{3} {\mathrm e}^{8 d x +8 c}-2100 a^{3} {\mathrm e}^{6 d x +6 c}-770 b^{3} {\mathrm e}^{6 d x +6 c}\right )}{105 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{7}}\)	\(425\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*coth(d*x+c)^2)^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-4*a*b^3*coth(d*x+c)-4*a^3*b*coth(d*x+c)-6*a^2*b^2*coth(d*x+c)+1/2*(a^4+4*a^3*b+6*a^2*b^2+4*a*b^3+b^4)*ln
(coth(d*x+c)+1)-4/5*a*b^3*coth(d*x+c)^5-2*a^2*b^2*coth(d*x+c)^3-4/3*a*b^3*coth(d*x+c)^3-b^4*coth(d*x+c)-1/7*b^
4*coth(d*x+c)^7-1/2*(a^4+4*a^3*b+6*a^2*b^2+4*a*b^3+b^4)*ln(coth(d*x+c)-1)-1/3*b^4*coth(d*x+c)^3-1/5*b^4*coth(d
*x+c)^5)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (104) = 208\).
time = 0.27, size = 410, normalized size = 3.73 \begin {gather*} \frac {1}{105} \, b^{4} {\left (105 \, x + \frac {105 \, c}{d} - \frac {8 \, {\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} - 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} - 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} - 105 \, e^{\left (-12 \, d x - 12 \, c\right )} - 44\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}}\right )} + \frac {4}{15} \, a b^{3} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} - 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} - 45 \, e^{\left (-8 \, d x - 8 \, c\right )} - 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + 2 \, a^{2} b^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + 4 \, a^{3} b {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + a^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(d*x+c)^2)^4,x, algorithm="maxima")

[Out]

1/105*b^4*(105*x + 105*c/d - 8*(203*e^(-2*d*x - 2*c) - 609*e^(-4*d*x - 4*c) + 770*e^(-6*d*x - 6*c) - 770*e^(-8
*d*x - 8*c) + 315*e^(-10*d*x - 10*c) - 105*e^(-12*d*x - 12*c) - 44)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*
c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 1
4*c) - 1))) + 4/15*a*b^3*(15*x + 15*c/d - 2*(70*e^(-2*d*x - 2*c) - 140*e^(-4*d*x - 4*c) + 90*e^(-6*d*x - 6*c)
- 45*e^(-8*d*x - 8*c) - 23)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x -
8*c) + e^(-10*d*x - 10*c) - 1))) + 2*a^2*b^2*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) - 2)/(
d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1))) + 4*a^3*b*(x + c/d + 2/(d*(e^(-2*d*x - 2*
c) - 1))) + a^4*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1164 vs. \(2 (104) = 208\).
time = 0.44, size = 1164, normalized size = 10.58 \begin {gather*} -\frac {4 \, {\left (105 \, a^{3} b + 210 \, a^{2} b^{2} + 161 \, a b^{3} + 44 \, b^{4}\right )} \cosh \left (d x + c\right )^{7} + 28 \, {\left (105 \, a^{3} b + 210 \, a^{2} b^{2} + 161 \, a b^{3} + 44 \, b^{4}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - {\left (420 \, a^{3} b + 840 \, a^{2} b^{2} + 644 \, a b^{3} + 176 \, b^{4} + 105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x\right )} \sinh \left (d x + c\right )^{7} - 28 \, {\left (75 \, a^{3} b + 120 \, a^{2} b^{2} + 71 \, a b^{3} + 14 \, b^{4}\right )} \cosh \left (d x + c\right )^{5} + 7 \, {\left (420 \, a^{3} b + 840 \, a^{2} b^{2} + 644 \, a b^{3} + 176 \, b^{4} + 105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x - 3 \, {\left (420 \, a^{3} b + 840 \, a^{2} b^{2} + 644 \, a b^{3} + 176 \, b^{4} + 105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 140 \, {\left ({\left (105 \, a^{3} b + 210 \, a^{2} b^{2} + 161 \, a b^{3} + 44 \, b^{4}\right )} \cosh \left (d x + c\right )^{3} - {\left (75 \, a^{3} b + 120 \, a^{2} b^{2} + 71 \, a b^{3} + 14 \, b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 84 \, {\left (45 \, a^{3} b + 60 \, a^{2} b^{2} + 41 \, a b^{3} + 14 \, b^{4}\right )} \cosh \left (d x + c\right )^{3} - 7 \, {\left (5 \, {\left (420 \, a^{3} b + 840 \, a^{2} b^{2} + 644 \, a b^{3} + 176 \, b^{4} + 105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )^{4} + 1260 \, a^{3} b + 2520 \, a^{2} b^{2} + 1932 \, a b^{3} + 528 \, b^{4} + 315 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x - 10 \, {\left (420 \, a^{3} b + 840 \, a^{2} b^{2} + 644 \, a b^{3} + 176 \, b^{4} + 105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 28 \, {\left (3 \, {\left (105 \, a^{3} b + 210 \, a^{2} b^{2} + 161 \, a b^{3} + 44 \, b^{4}\right )} \cosh \left (d x + c\right )^{5} - 10 \, {\left (75 \, a^{3} b + 120 \, a^{2} b^{2} + 71 \, a b^{3} + 14 \, b^{4}\right )} \cosh \left (d x + c\right )^{3} + 9 \, {\left (45 \, a^{3} b + 60 \, a^{2} b^{2} + 41 \, a b^{3} + 14 \, b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 420 \, {\left (5 \, a^{3} b + 6 \, a^{2} b^{2} + 5 \, a b^{3}\right )} \cosh \left (d x + c\right ) - 7 \, {\left ({\left (420 \, a^{3} b + 840 \, a^{2} b^{2} + 644 \, a b^{3} + 176 \, b^{4} + 105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )^{6} - 5 \, {\left (420 \, a^{3} b + 840 \, a^{2} b^{2} + 644 \, a b^{3} + 176 \, b^{4} + 105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )^{4} - 2100 \, a^{3} b - 4200 \, a^{2} b^{2} - 3220 \, a b^{3} - 880 \, b^{4} - 525 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x + 9 \, {\left (420 \, a^{3} b + 840 \, a^{2} b^{2} + 644 \, a b^{3} + 176 \, b^{4} + 105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{105 \, {\left (d \sinh \left (d x + c\right )^{7} + 7 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{5} + 7 \, {\left (5 \, d \cosh \left (d x + c\right )^{4} - 10 \, d \cosh \left (d x + c\right )^{2} + 3 \, d\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (d \cosh \left (d x + c\right )^{6} - 5 \, d \cosh \left (d x + c\right )^{4} + 9 \, d \cosh \left (d x + c\right )^{2} - 5 \, d\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(d*x+c)^2)^4,x, algorithm="fricas")

[Out]

-1/105*(4*(105*a^3*b + 210*a^2*b^2 + 161*a*b^3 + 44*b^4)*cosh(d*x + c)^7 + 28*(105*a^3*b + 210*a^2*b^2 + 161*a
*b^3 + 44*b^4)*cosh(d*x + c)*sinh(d*x + c)^6 - (420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a
^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x)*sinh(d*x + c)^7 - 28*(75*a^3*b + 120*a^2*b^2 + 71*a*b^3 + 14*b^4)*cosh(
d*x + c)^5 + 7*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4
)*d*x - 3*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x
)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 140*((105*a^3*b + 210*a^2*b^2 + 161*a*b^3 + 44*b^4)*cosh(d*x + c)^3 - (75
*a^3*b + 120*a^2*b^2 + 71*a*b^3 + 14*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + 84*(45*a^3*b + 60*a^2*b^2 + 41*a*b^
3 + 14*b^4)*cosh(d*x + c)^3 - 7*(5*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2
*b^2 + 4*a*b^3 + b^4)*d*x)*cosh(d*x + c)^4 + 1260*a^3*b + 2520*a^2*b^2 + 1932*a*b^3 + 528*b^4 + 315*(a^4 + 4*a
^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x - 10*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b
+ 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 28*(3*(105*a^3*b + 210*a^2*b^2 + 161*a*b
^3 + 44*b^4)*cosh(d*x + c)^5 - 10*(75*a^3*b + 120*a^2*b^2 + 71*a*b^3 + 14*b^4)*cosh(d*x + c)^3 + 9*(45*a^3*b +
60*a^2*b^2 + 41*a*b^3 + 14*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 - 420*(5*a^3*b + 6*a^2*b^2 + 5*a*b^3)*cosh(d*x
+ c) - 7*((420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*
x)*cosh(d*x + c)^6 - 5*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b
^3 + b^4)*d*x)*cosh(d*x + c)^4 - 2100*a^3*b - 4200*a^2*b^2 - 3220*a*b^3 - 880*b^4 - 525*(a^4 + 4*a^3*b + 6*a^2
*b^2 + 4*a*b^3 + b^4)*d*x + 9*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2
+ 4*a*b^3 + b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^7 + 7*(3*d*cosh(d*x + c)^2 - d)*sinh(d*
x + c)^5 + 7*(5*d*cosh(d*x + c)^4 - 10*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^3 + 7*(d*cosh(d*x + c)^6 - 5*d*c
osh(d*x + c)^4 + 9*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (99) = 198\).
time = 8.32, size = 522, normalized size = 4.75 \begin {gather*} \begin {cases} - \frac {a^{4} \log {\left (- e^{- d x} \right )}}{d} - \frac {4 a^{3} b \log {\left (- e^{- d x} \right )} \coth ^{2}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} - \frac {6 a^{2} b^{2} \log {\left (- e^{- d x} \right )} \coth ^{4}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} - \frac {4 a b^{3} \log {\left (- e^{- d x} \right )} \coth ^{6}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} - \frac {b^{4} \log {\left (- e^{- d x} \right )} \coth ^{8}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} & \text {for}\: c = \log {\left (- e^{- d x} \right )} \\- \frac {a^{4} \log {\left (e^{- d x} \right )}}{d} - \frac {4 a^{3} b \log {\left (e^{- d x} \right )} \coth ^{2}{\left (d x + \log {\left (e^{- d x} \right )} \right )}}{d} - \frac {6 a^{2} b^{2} \log {\left (e^{- d x} \right )} \coth ^{4}{\left (d x + \log {\left (e^{- d x} \right )} \right )}}{d} - \frac {4 a b^{3} \log {\left (e^{- d x} \right )} \coth ^{6}{\left (d x + \log {\left (e^{- d x} \right )} \right )}}{d} - \frac {b^{4} \log {\left (e^{- d x} \right )} \coth ^{8}{\left (d x + \log {\left (e^{- d x} \right )} \right )}}{d} & \text {for}\: c = \log {\left (e^{- d x} \right )} \\x \left (a + b \coth ^{2}{\left (c \right )}\right )^{4} & \text {for}\: d = 0 \\a^{4} x + 4 a^{3} b x - \frac {4 a^{3} b}{d \tanh {\left (c + d x \right )}} + 6 a^{2} b^{2} x - \frac {6 a^{2} b^{2}}{d \tanh {\left (c + d x \right )}} - \frac {2 a^{2} b^{2}}{d \tanh ^{3}{\left (c + d x \right )}} + 4 a b^{3} x - \frac {4 a b^{3}}{d \tanh {\left (c + d x \right )}} - \frac {4 a b^{3}}{3 d \tanh ^{3}{\left (c + d x \right )}} - \frac {4 a b^{3}}{5 d \tanh ^{5}{\left (c + d x \right )}} + b^{4} x - \frac {b^{4}}{d \tanh {\left (c + d x \right )}} - \frac {b^{4}}{3 d \tanh ^{3}{\left (c + d x \right )}} - \frac {b^{4}}{5 d \tanh ^{5}{\left (c + d x \right )}} - \frac {b^{4}}{7 d \tanh ^{7}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(d*x+c)**2)**4,x)

[Out]

Piecewise((-a**4*log(-exp(-d*x))/d - 4*a**3*b*log(-exp(-d*x))*coth(d*x + log(-exp(-d*x)))**2/d - 6*a**2*b**2*l
og(-exp(-d*x))*coth(d*x + log(-exp(-d*x)))**4/d - 4*a*b**3*log(-exp(-d*x))*coth(d*x + log(-exp(-d*x)))**6/d -
b**4*log(-exp(-d*x))*coth(d*x + log(-exp(-d*x)))**8/d, Eq(c, log(-exp(-d*x)))), (-a**4*log(exp(-d*x))/d - 4*a*
*3*b*log(exp(-d*x))*coth(d*x + log(exp(-d*x)))**2/d - 6*a**2*b**2*log(exp(-d*x))*coth(d*x + log(exp(-d*x)))**4
/d - 4*a*b**3*log(exp(-d*x))*coth(d*x + log(exp(-d*x)))**6/d - b**4*log(exp(-d*x))*coth(d*x + log(exp(-d*x)))*
*8/d, Eq(c, log(exp(-d*x)))), (x*(a + b*coth(c)**2)**4, Eq(d, 0)), (a**4*x + 4*a**3*b*x - 4*a**3*b/(d*tanh(c +
d*x)) + 6*a**2*b**2*x - 6*a**2*b**2/(d*tanh(c + d*x)) - 2*a**2*b**2/(d*tanh(c + d*x)**3) + 4*a*b**3*x - 4*a*b
**3/(d*tanh(c + d*x)) - 4*a*b**3/(3*d*tanh(c + d*x)**3) - 4*a*b**3/(5*d*tanh(c + d*x)**5) + b**4*x - b**4/(d*t
anh(c + d*x)) - b**4/(3*d*tanh(c + d*x)**3) - b**4/(5*d*tanh(c + d*x)**5) - b**4/(7*d*tanh(c + d*x)**7), True)
)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (104) = 208\).
time = 0.43, size = 447, normalized size = 4.06 \begin {gather*} \frac {105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (d x + c\right )} - \frac {8 \, {\left (105 \, a^{3} b e^{\left (12 \, d x + 12 \, c\right )} + 315 \, a^{2} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 315 \, a b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 105 \, b^{4} e^{\left (12 \, d x + 12 \, c\right )} - 630 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} - 1575 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 1260 \, a b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 315 \, b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} + 3360 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2555 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 770 \, b^{4} e^{\left (8 \, d x + 8 \, c\right )} - 2100 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - 3990 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3080 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 770 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 2835 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2121 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 609 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} - 630 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} - 1155 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 812 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 203 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} b + 210 \, a^{2} b^{2} + 161 \, a b^{3} + 44 \, b^{4}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{7}}}{105 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(d*x+c)^2)^4,x, algorithm="giac")

[Out]

1/105*(105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(d*x + c) - 8*(105*a^3*b*e^(12*d*x + 12*c) + 315*a^2*b^
2*e^(12*d*x + 12*c) + 315*a*b^3*e^(12*d*x + 12*c) + 105*b^4*e^(12*d*x + 12*c) - 630*a^3*b*e^(10*d*x + 10*c) -
1575*a^2*b^2*e^(10*d*x + 10*c) - 1260*a*b^3*e^(10*d*x + 10*c) - 315*b^4*e^(10*d*x + 10*c) + 1575*a^3*b*e^(8*d*
x + 8*c) + 3360*a^2*b^2*e^(8*d*x + 8*c) + 2555*a*b^3*e^(8*d*x + 8*c) + 770*b^4*e^(8*d*x + 8*c) - 2100*a^3*b*e^
(6*d*x + 6*c) - 3990*a^2*b^2*e^(6*d*x + 6*c) - 3080*a*b^3*e^(6*d*x + 6*c) - 770*b^4*e^(6*d*x + 6*c) + 1575*a^3
*b*e^(4*d*x + 4*c) + 2835*a^2*b^2*e^(4*d*x + 4*c) + 2121*a*b^3*e^(4*d*x + 4*c) + 609*b^4*e^(4*d*x + 4*c) - 630
*a^3*b*e^(2*d*x + 2*c) - 1155*a^2*b^2*e^(2*d*x + 2*c) - 812*a*b^3*e^(2*d*x + 2*c) - 203*b^4*e^(2*d*x + 2*c) +
105*a^3*b + 210*a^2*b^2 + 161*a*b^3 + 44*b^4)/(e^(2*d*x + 2*c) - 1)^7)/d

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Mupad [B]
time = 1.31, size = 111, normalized size = 1.01 \begin {gather*} x\,{\left (a+b\right )}^4-\frac {{\mathrm {coth}\left (c+d\,x\right )}^3\,\left (6\,a^2\,b^2+4\,a\,b^3+b^4\right )}{3\,d}-\frac {{\mathrm {coth}\left (c+d\,x\right )}^5\,\left (b^4+4\,a\,b^3\right )}{5\,d}-\frac {b^4\,{\mathrm {coth}\left (c+d\,x\right )}^7}{7\,d}-\frac {b\,\mathrm {coth}\left (c+d\,x\right )\,\left (4\,a^3+6\,a^2\,b+4\,a\,b^2+b^3\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*coth(c + d*x)^2)^4,x)

[Out]

x*(a + b)^4 - (coth(c + d*x)^3*(4*a*b^3 + b^4 + 6*a^2*b^2))/(3*d) - (coth(c + d*x)^5*(4*a*b^3 + b^4))/(5*d) -
(b^4*coth(c + d*x)^7)/(7*d) - (b*coth(c + d*x)*(4*a*b^2 + 6*a^2*b + 4*a^3 + b^3))/d

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